I have been pondering the issue of what makes a theorem noteworthy. There are many famous examples of 'outstanding' theorems, such as Roth's theorem in Diophantine approximation, Szemeredi's Theorem, Vinogradov's Theorem, etc. There are also examples of theorems that are only marginal improvements despite significant effort... such as Dyson's theorem (which only improved Siegel's theorem from $2\sqrt{d}$ to $\sqrt{2d}$, where $d$ is the degree of the algebraic number being approximated). In fact I was told that Dyson gave up mathematics because he worked so hard only to get what he thought was a very small improvement.

So what do you think is a 'good' theorem? Presumably when you set out to work out a theorem, you intend your theorem to be good. So how do you decide what is a good theorem to prove? How do you decide which results are important achievements that are noteworthy?

Some of my criteria include:

1) The perceived strength of the hypothesis and the conclusion. For example, the following theorem is surely not good: Suppose every element of a set $A$ is a power of 2 larger than 1. Then every element of $A$ is even. In this case the hypothesis is extremely strong, and the conclusion extremely weak.

However, here's an example of what I consider to be an outstanding theorem, based solely on the weakness of the hypothesis and the strength of the result: Suppose $\alpha$ is an algebraic number of degree $d$, and suppose that $\epsilon > 0$. Then there exists only finitely many integers $p,q$ with $\gcd(p,q) = 1$ such that $$\displaystyle \left | \alpha - \frac{p}{q} \right| < \frac{1}{q^{2 + \epsilon}}$$

Here all we are assuming that $\alpha$ is algebraic, which is not a particularly strong assertion... but the conclusion is very strong. This is compared to Thue's theorem, Siegel's theorem, and Dyson's theorem where $2 + \epsilon$ is replaced with $d/2 + \epsilon$, $2\sqrt{d} + \epsilon$, and $\sqrt{2d} + \epsilon$ respectively.

2) How much it improved on the previous best result. For example, Szmeredi's theorem which asserted that the assumption of a subset $A \subset \mathbb{N}$ having positive upper density is sufficient for $A$ to have arithmetic progressions of length $k$ for any $k \geq 3$ is a significant improvement over Roth's (other) theorem which asserted the same for length three arithmetic progressions.

3) How much it spurred further research. Examples in this category could include the $h$-cobordism theorem, Falting's theorem, and Roth's theorem.

So what qualities do you think make up a magnificent theorem and what are some examples?

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The question in the last sentence of this question is way too broad. At the very least, the question needs to be tightened up to ask about qualities that make a theorem "good" (not just a laundry list of people's favorite theorems) and be community wiki. But I suspect that even a tightened up question will be closed.
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Andy PutmanJun 20 '11 at 17:02

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I'm not voting to close... but I'm very tempted. And I agree with Andy: this should be community-wiki.
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André HenriquesJun 20 '11 at 17:05

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This is not a mathematical question, just a collection of statements. Voted to close. In any case if somebody finds it interesting, it should be a community wiki.
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Mark SapirJun 20 '11 at 17:16

I think this question is very interesting, when narrowed to the perspective of young-career mathematicians trying to decide what to work on. (Presumably Roth, Szemeredi, et al. did not have to worry about this!)
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Frank ThorneJun 20 '11 at 17:31